As discussed in Ref. [15] the elastic quantities (stresses and displacements) satisfy a fourth-order system of ordinary differential equations. We first introduce the stress-displacement vector, defined by

where **u** is the horizontal displacement, **w** is the vertical
displacement, is the tangential stress and is
the normal stress. The purpose of introducing the scaling of **u** and
given in Eq. () is to
eliminate complex quantities from the governing equations and to obtain
a form where the eigenvalue **k** occurs only in squared form. The
stress-displacement vector then satisfies

where,

where the quantities and are defined by

and denote the **P** and **S** wave velocities respectively. In
this form certain properties of elastic waves are immediately obvious.
For instance, since the eigenvalue occurs only as a squared quantity the
the eigenvalues will come in pairs. That is, if is an
eigensolution then is also an eigensolution.

The above equations for are combined with interfacial and boundary conditions to completely specify the acousto-elastic modal problem. At an elastic-elastic interface, one requires continuity of (i.e., continuity of displacements and stresses). At an acousto-elastic interface the condition of continuity of horizontal displacement is relaxed. Noting that pressure is the negative of the normal stress, , vanishes in an acoustic medium, and the gradient of the pressure gives the time derivative of the velocity field, one obtains,

* KRAKEN*
and * KRAKENC*
use the reduced delta-matrix formulation. This is
obtained by introducing a new set of dependent variables defined by,

where and denote two linearly independent solutions in the elastic medium. Note that involves all permutations of with an ordering chosen to obtain a simple form for the equations. By differentiating the above equations and substituting into Eq. () we find that satisfies a system of differential equations:

where,

The differential equation for reduces to and has
been eliminated from the system. In terms of the **y**-functions the
interface conditions between the acoustic medium and a stratified
elastic bottom can be written as,

with,

Tue Oct 28 13:27:38 PST 1997